whenever smooth vertex is pressed it breaks up mirrored objects
You are doing this:

Cutting a cube and sticking a Mirror Modifier on it.



Then doing Ctrl+V, and picking 'Smooth Vertex'



This produces exactly what you show, and that's exactly what we would expect.



Smooth Vertex tries to pull all real vertices together. As mentioned in the docs 


  This will apply once the Smooth Tool.


You must remember you are performing this Vertex operation on only half a cube, (the rest is mirrored) and therefor the real center of all these real vertices (shaded in orange) lies within their collective coordinates, not on the mirror axis - the mirrored portion is not taken into account.

Objects with a bit more definition tend to smooth much more subtle, retaining much of the silhouette and scale.



I think your main confusion was that you where performing this operation on an object that relies on a Mirror Modifier for half of it's visual geometry. Maybe you are looking for the Laplacian Smooth deform modifier. (which will work ontop of the Mirror Modifier)



Or perhaps the Subdivision Surface Modifier.